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Rough differential equations driven by TFBM with Hurst index $H\in (\frac{1}{4}, \frac{1}{3})$

Lijuan Zhang, Jianhua Huang

Abstract

We consider the rough differential equations driven by tempered fractional Brownian motion with Hurst index $H\in (\frac{1}{4}, \frac{1}{3})$ and tempered parameter $λ>0$. First, by means of piecewise linear approximation, we canonically lift the tempered fractional Brownian motion to a three-step geometric rough path in an almost sure sense. Subsequently, employing the Doss-Sussmann technique in conjunction with a greedy sequence of stopping times, we construct a suitable transformation that establishes a bijection between the solution of the rough differential equation and that of an associated ordinary differential equation. This yields the existence and uniqueness of a solution to the original equation. Based on this result and appealing to Gronwall's lemma, we further derive an upper bound for the solution norm, thereby providing a quantitative control on its growth.

Rough differential equations driven by TFBM with Hurst index $H\in (\frac{1}{4}, \frac{1}{3})$

Abstract

We consider the rough differential equations driven by tempered fractional Brownian motion with Hurst index and tempered parameter . First, by means of piecewise linear approximation, we canonically lift the tempered fractional Brownian motion to a three-step geometric rough path in an almost sure sense. Subsequently, employing the Doss-Sussmann technique in conjunction with a greedy sequence of stopping times, we construct a suitable transformation that establishes a bijection between the solution of the rough differential equation and that of an associated ordinary differential equation. This yields the existence and uniqueness of a solution to the original equation. Based on this result and appealing to Gronwall's lemma, we further derive an upper bound for the solution norm, thereby providing a quantitative control on its growth.
Paper Structure (9 sections, 14 theorems, 251 equations)

This paper contains 9 sections, 14 theorems, 251 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. TFBM
  4. Rough paths
  5. Rough integrals
  6. Greedy sequence of stopping times
  7. Geometric rough path over TFBM
  8. Existence-uniqueness theorem and solution norm estimates
  9. Appendix

Key Result

Lemma 3.1

For any $m, n \geq 1$, $k = 1, 2, \cdots, 2^n$, $H\in (\frac{1}{4}, \frac{1}{2}]$, $\lambda>0$ and $p\in (2,4)$, the following estimate holds at the first level paths: where $c$ is some positive constant not depending on $m, n, k$.

Theorems & Definitions (25)

  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Proposition 3.4
  • Lemma 3.5
  • proof
  • Theorem 3.6
  • ...and 15 more